Problem: The Gades Foundation and HBM Corporation are eager to donate generously to their favorite charities. The Gades Foundation donated $\$250$ in the first month, and each month afterward their cumulative donation total increases by $\$100$. The HBM Corporation donated $\$64$ in the first month, and each month afterward their cumulative donation total increases by $75\%$. They started making donations at the same time, and they both make their monthly donations at the beginning of each month. What is the first month in which HBM Corporation's cumulative donation total exceeds the Gades Foundations' cumulative total?
Solution: Notice that the Gades Foundation's total grows linearly while the HBM Corporation's total grows exponentially. This means the HBM Corporation's total is bound to exceed the Gades Foundation's total at some point. Let's start calculating each organization's total cumulative donation to see when that happens. Month Gades Foundation HBM Corporation (Add $100$ each month.) (Multiply by $1.75$ each month.) $1$ $250$ $64$ $2$ $350$ $112$ $3$ $450$ $196$ $4$ $550$ $343$ $5$ $650$ $600.25$ $6$ $750$ $1050.44$ In conclusion, the HBM Corporation's cumulative donation total will first exceed the Gades Foundations' cumulative donation total in month number $6$.